Summary of the Phd thesis Theoretical and mathematical physics: The 3-3-1 simple model and the 3-2-2-1 model for dark matter and neutrino masses

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The purposes of research: Searching of dark matter in the proposed model called the simple 3-3-1 model (S331M); solving the neutrino mass problem and determine the Higgs spectrum in the G221 model with lepton-flavor non-universality.

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Nội dung Text: Summary of the Phd thesis Theoretical and mathematical physics: The 3-3-1 simple model and the 3-2-2-1 model for dark matter and neutrino masses

MINISTRY OF EDUCATION VIETNAM ACADEMY OF SCIENCE AND TRAINING AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ……..….***………… NGUYEN THI KIM NGAN SIMPLE 3-3-1 MODEL AND 3-2-2-1 MODEL FOR DARK MATTER AND NEUTRINO MASSES Speciality: Theoretical and mathematical physics Code: 62 44 01 03 SUMMARY OF THE PHD THESIS Hanoi – 2018
This thesis was completed at Graduate University of Science and Technology, Vietnam Academy of Science and Technology. Supervisors: Dr. Phung Van Dong Prof. Hoang Ngoc Long Referee 1: Prof. Dang Van Soa Referee 2: Dr. Dinh Nguyen Dinh Referee 3: Dr. Tran Minh Hieu This dissertation will be defended in front of the evaluating assembly at academy level, place of defending: meeting room, Graduate University of Science and Technology, Vietnam Academy of Science and Technology. This thesis can be studied at: - The Library of Graduate University of Science and Technology - The Vietnam National Library
1 INTRODUCTION The Standard Model (SM) has been successful in exactly pre- dicting many observational experimental results. Successes of SM can be mentioned such as predicting the Z and W boson, gluons, c (charm) quark, t (top) quark and b (bottom) quark before they were observed by the experiments. One of those is prediction of Higgs boson recently discovered by LHC (Large Hadron Collider) at CERN with the 125 GeV mass. This is the last particle predicted by SM. However, to this day there are much experimental data re- maining beyond prediction of SM, such as: • Why does t (top) quark have the uncommon heavy mass? SM predicted t quark has the approximate 10 GeV mass while the experimental result of Tevatron at Fermilab in 1995 demon- strated that t quark has the 173 GeV mass. • The early universe is a quantum system, therefore the number of particles equals to the one of anti-particles, why the present universe only includes matter constituted by particles, there is no evidence for the existence of antimatter structured by anti-particles, called matter-antimatter asymmetry or baryon asymmetry. • SM predicted neutrinos have zero masses because they do not have the right-handed components and lepton number is conserved. However, the solar, atmospheric, accelerator and reactor neutrino experiments have predicated in most of 20 years that there are neutrino ocillations when they propa- gate a long enough journey. This requires neutrinos to have nonzero masses (even if they are smaller than 1 eV) and mix- ing. There are three flavours of neutrinos and their mixing parameterise through the three Euler angles and three CP vi- olation phases (1 Dirac phase and 2 Majorana phases). The
2 existing data of the recent experiments have showed that the squared mass differences and the mixing angles of neutrinos have their defined values. There is large mixing between the electron neutrino and the muon neutrino, between the muon neutrino and the tau neutrino, while there is small mixing (different to zero) between the electron neutrino and the tau neutrino. This is completely different from the quark mixing (all of them are small). The neutrino experiments can just determine the Dirac phase which can be different to zero and can not define the Majorana phases. Then, are the neutrinos Dirac or Majorana fermions? How can generate the naturally small neutrino masses which are appropriate for the exper- imental data? why does the flavour mixing of quarks and leptons have the completely determined mixing angles? If there is the existence of right-handed neutrinos νaR they are colorless, null isospin and null weak hypercharge. Thus, they do not have gauge interactions, called sterile particles. How- ever, they can be meaningful in generating neutrino masses and in the baryon number asymmetry of the universe. In fact, when νaR is added neutrinos can get Dirac masses because of the interaction with the Higgs boson, mD ∼ v (electroweak scale), which is similar to the charged fermions. Because νaR is the singlet of SM they can get large Majorana masses, mR , which violate lepton number. As a result, the active neutri- nos ∼ νaL gain Majorana masses by the seesaw mechanism, mL = −(mD )2 /mR , which is naturally small because of the condition mR mD . It is similar to The Grand Unified The- ory (GUT) SO(10) that the Dirac masses are proportional to the electroweak scale, mD ∼ 100 GeV. The active neutrinos mL ∼ eV , then mR ∼ 1013 GeV is in the GUT scale and this is a motivation of the GUT, SO(10). However, the GUT is difficult to observe by the experiments and encounters the problem of unnatural hierarchy. The idea of the GUT can be rejected and mR is imposed so that mR ∼ TeV, this scale is discovered by LHC, then the value of mD is approximately the electron mass. We have the seesaw mechanism at TeV scale. Yet, a new problem arises is that what is the nature of right-handed neutrinos (νaR )? • One of the problems has recently attracted to the theoreti- cal and experimental physicists is that the existence of the
3 amount of unobserved matter (Dark Matter - DM). This day, there are two viewpoints about DM. Those are the baryonic DM and non-baryonic DM. The candidates of baryonic DM are neutron stars or black holes which is a research field of astrophysics and cosmology, while the ones of non-baryonic DM are WIMPs (Weakly Interacting Massive Particles) which are massive particles, and interact very weakly with normal particles. WIMPs are the objects searched by the elementary particle physicists. From the point of view of particle physics, a DM particle must be an electrically neutral particle, stable and satisfy the relic density of DM. Although WIMPs have been still found at the colliders, a variety of evidence from as- trophysics and cosmology has confirmed the existence of DM in the recent decades. The typical astrophysical evidence in recent data from the Planck’s satellite shows that the amount of non-baryonic DM in the universe accounts for 26.8%, which is different to 23% of the previous WMAP data. In fact, SM is proven to contain any particle that is a candidate for DM. • An another current noticeable problem for theoretical physi- cists is the experimental signal obtained in 2014 at LHCb on B meson anomaly decays with 3.5 σ in comparison to the SM prediction. This shows there is the violation of the lep- ton flavour universality or in other words there is the lepton flavour non-universality (LNU) that is different to the lepton flavour universality in SM. For these reasons, we find that SM is not a complete theory for particle physics, so SM need to be expanded. Now, a physical model must satisfy the following requirements: i) At low energy (about 200 GeV), the model must include SM. ii) The neutrino masses and mixing angles consistent with the neutrino oscillation experiments. iii) Explain the baryon asymmetry of the Universe (BAU). iv) Higgs spectrum consistent with current data Higgs, contain the SM Higgs-like particle which has characteristics similar to the Higgs of SM. v) Contain the new particles which are the candidates for DM. However, new physical models are built initially so that they satisfy the certain aforementioned requirements and some experimental results which recently discovered. These models will continue to be completed progressively to fully explain the existing experimental results. In the current expanded SM models, the experimental data on neutrino oscillations and DM are the
4 first requirement to satisfy. Therefore, the study of the viability of the candidate of the DM or / and the appropriate experimental neutrino oscillation to consider the practicality of the new models is interesting issue, and not less important. Theoretically to obtain the candidate of DM with the relic density which fits the existing experimental data, the extended SM model must contain electrically neutral and stable particles, whereby they do not decay quickly into the SM particles. Namely, the channels of a DM particle into two the SM particles must be very small, and consequently the corresponding triple interaction coefficients have very small or zero. To completely eliminate the triple interactions, the theory is assigned to preserve under the Z2 symmetry, whereby the model will contain the SM particles which are always even under the Z2 , and contains at least one new elec- trically neutral particle which is odd under this symmetry. The theory, always preserve the Z2 , would eliminate the triple inter- actions containing the new neutral particle and two SM particles. Then, the DM decay channels only remain the channels which two DM particles annihilate each other to create two SM particles. The models extended from SM with the DM aforementioned can- didates have been studied extensively and has explained very well the experimental data on DM. However, the DM light candidates predicted by supersymmetry, extra dimensions, dark photon and the extensions of SM with inert particles has been eliminated by LHC and the other experiments. The search for a DM candidate in this direction for a class of extended models which has non-Abel gauge symmetry such as the 3-3-1 models has just recently been researched on some certain models. In particular, the reduced minimal 3-3-1 model (RM331M), the reduced version of the minimal 3-3-1 model (M331M), has a simple Higgs spectrum but it still many limitations: the FC- NCs are too large, the ρ parameter is not suitable the experimen- tal data, and it does not contain DM candidates. In this thesis, we solved the above problems on the basis of constructing a new model, called a simple 3-3-1 model, by properly rearranging the fermion spectrum and choosing the new Higgs spectrum that is different from RM331M. In addition, the DM problem is solved by assuming the model conserving the Z2 symmetric and containing the inert scalar multiplets (odd under the Z2 ) on the TeV scale (the 3-3-1 breaking scale).
5 The new components in these multiplets will give the small- est physical particle as the DM candidate, since it always has an odd charge of the Z2 , so it does not contain the triple interactions (the interaction of one new neutral particle and two SM particles). The possibilities for specific relic densities according to the decay channel of two DM particles depend on the representation of the inert scalars and need to be investigated specifically to compare with the experiments. We review some of the features and history of the simple 3-3-1 model: • Our model is based on the minimal 3-3-1 model (M331M). The gauge symmetry group of this model is SU (3)C ⊗SU (3)L ⊗ U (1)X , where the color group SU (3)C is as usual while the two last groups was enlarged from SU (2)L ⊗ U (1)Y (the elec- troweak group of SM). M331M has the leptons that is used similar to the leptons of SM, while the third generation of quarks transforms under SU (3)L differently from the two oth- ers. Because the M331M has traditionally been studied to be worked with three scalar triplets ρ = (ρ+ 0 ++ 1 , ρ2 , ρ3 ), η = 0 − + − −− 0 (η1 , η2 , η3 ), χ = (χ1 , χ2 , χ3 ) and/nor one scalar sextet 0 − + −− 0 ++ S = (S11 , S12 , S13 , S22 , S23 , S33 ).Therefore, the Higgs sector of M331M is complex, no solution, and the model does not give DM candidates. • Although the M331M has been researched to deal with the complex Higgs sector through RM331M, this model still has problems that are not suitable for the experimental data as mentioned above. • To overcome the disadvantages of M331M and RM331M, we have proposed the simple 3-3-1 model (S331M) that has the same particle structure of RM331M. However, S331M differs from RM331M as follows: the second and third quark genera- tions transform as the antitriplets under SU (3)L in RM331M, while in our model the first and second quark generations transform as the triplets. Both models have two scalar triplets but there is a difference in their scalar sectors. Namely, two scalar triplets rho and chi are used in RM331M, while eta and chi are used in S331M. S331M with the particle ar- rangement as chosen will result in a suitable small FCNC, the parameter rho having the appropriate value, the t quark
6 receiving the proper mass at tree level, and especially the model will be give the candidates for DM by adding the inert scalars (odd under Z2 ) containing the DM candidate. The neutrino mass is explained by the approximate symmetry of B-L. In addition to the above-mentioned class of 3-3-1 models, the recent studies also focus on the new model in which the electroweak group of SM SU (2) otimesU (1)Y is expanded into SU (2)1 ⊗SU (2)2 ⊗ U (1)Y , and simultaneously adding vector-like fermions to reason- ably explain the recent experimental results of the anomaly in the decay processes of B meson, while the color group remains the same, so the new model is called G221. As we all know, the Glashow-Weinberg-Salam model (GWS), the model unifies the weak electromagnetic interaction and weak interaction, also called SM mentioned above, has a notable char- acteristic that is the identity of three quark/ lepton generations. This means the physics of the generations is identical, then we can consider the interactions of only one generation (example the first generation), the interactions of other generations can be found out similarly. However, the experimental data of B meson decay anomalies have shown: Γ(B¯ → D∗ τ ν˜) RD∗ = Γ(B ¯ → D∗ l ν˜) = 0.310 ± 0.015 ± 0.008 , Γ(B¯ → D τ ν˜) RD = Γ(B ¯ → D l ν˜) = 0.403 ± 0.040 ± 0.024, l = e, µ, (1) with 3.5 σ in comparison to the SM prediction: RD∗ = 0.252 ± 0.004, RD = 0.305 ± 0.012. (2) The above results provide hints for violation of the lepton flavor universality (LFU). Thus, a series of expanded models of MHC was developed in 2016 to further explain the newly published experimental results of the B meson decay anomalies. As mentioned above, these models must also fully explain the data of neutrinos and DM. One of those models is the G221. This model appropriately explains all data of the meson B decay anomalies. The G221 model contains the electroweak groups SU (2)1 and SU (2)2 in turn breaking at the high energy scales and then
7 at the electroweak scale of SM. The initial fermions in the model included light fermions with left- and right-handed components transform as the singlets under the SU (2)1 , while these left- and right-handed components respectively transform as the doublets and singlets under the SU (2)2 , which is similar to SM. The heavy fermions with the left- and right-handed components are added in this model which they transform as singlets under the SU (2)2 , dou- blets under the SU (2)1 , so called vector-like fermions. Beside one Higgs doublet of the SU (2)2 as in SM, the model also has one dou- blet of the SU (2)1 and one self-dual bidoublet of both SU (2)1 and SU (2)2 . Yukawa interactions between fermions and Higgs bosons give the fairly complex mass matrices of leptons and quarks. The consequence is the mixing matrices of fermion states give differ- ent mixing angles for different fermion generations. Thus, these physical fermions will interact differently with the gauge bosons in the model, thereby explaining the experimental results of the B meson decay. However, the neutrino mixing matrix in this model always gives the zero mass eigenvalue of light neutrinos, which is in conflict with the experimental results of neutrino oscillations. The above model also considers the Z2 symmetry, but this symmetry is softly-broken to ensure generating the right masses for the charged scalars in the model. Therefore, the model also does not contain the DM candidate. However, the possibility of DM in some of its extensions will be discussed. In order to solve the problem of neutrino mass in G221, we have found that the model contains two Higgs doublets, which are well suitable to the mechanism of neutrino mass generation as in the Zee model. An important thing is that when we apply the mechanism of neutrino mass generation as in the Zee model we only need to add a pair of singly charged scalars so that there is no new symmetry breaking scale. Thus, in addition to the neutrinos have very small masses produced by radiative corrections, all the mass, the mixing matrix, and the physical eigenstates of remaining particles are unaffected. Because of the current and urgent of the issues presented above, I chose emph “ The 3-3-1 simple model and the 3-2-2-1 model for dark matter and neutrino masses ”. The thesis focuses on two major issues: searching DM and generating of neutrino masses in the simple 3-3-1 model and the 3-2-2-1 model.
8 The purposes of research: • Searching of dark matter in the proposed model called the simple 3-3-1 model (S331M). • Solving the neutrino mass problem and determine the Higgs spec- trum in the G221 model with lepton-flavor non-universality. The objects of research: • The candidates for dark Matter in the simple 3-3-1 model. • The neutrino masses and Higgs spectrum in the G221 model with lepton-flavor non-universality. The contents of research: • Propose the simple 3-3-1 model. • Introduce the inert scalar multiplets to SM331M for searching of DM candidates. • Generate neutrino masses from one loop corrections and study details the characteristics of the Higgs spectrum in the G221 model with lepton-flavor non-universality (LNU). The methods of research: • Quantum field theory. • The Feynman rule to calculate the amplitudes and the decay widths. • The Wolfram Mathematica software to solve the numerical cal- culation and to process the complex analytic reduction in the de- termination of interactive coefficients. The structure of thesis: In this thesis, apart from the introduction, conclusions and appen- dices, the main content of the thesis is presented in three chapters (according to the list of publications, Chapter 1 and Chapter 2 have the content of the first publication, Chapter 3 has the content of the third publication):
9 Chapter 1: We constructed the simple 3-3-1 model, in which we have arranged the particle structure in the model, defined the physical Higgs scalar fields, the gauge bosons, the fermion masses and the proton stability as well as calculated the FCNCs. Chapter 2: We introduced the inert scalar multiplets to S331M for finding the DM candidates. Considering alternately S331M with the rho inert triplet and the S inert sextet as well as the replications of η and χ helps us identify the candidates for DM. We also made an estimate of the DM observations at the end of this chapter. Chapter 3: We abstracted the model based on the standard symmetry group SU(2)1 ⊗ SU(2)2 ⊗ U(1)Y , in which we solved the problem of generating active neutrino masses from loop corrections and studied the detailed features of the gauge boson and Higgs sections in this model. In order to see the detailed features of S331M, we go into the contents of Chapter 1.
10 Chapter 1 The simple 3-3-1 model Based on the reduced minimal 3-3-1 model and the minimal 3-3-1 model we will build a new model with the minimal lepton and scalar sector—called the simple 3-3-1 model. The model has showed the experimental fit. 1.1 The particle structure of the model The fermion content which is anomaly free is defined as:   νaL ψaL ≡  eaL  ∼ (1, 3, 0), (eaR )c   dαL QαL ≡  −uαL  ∼ (3, 3∗ , −1/3), JαL   u3L Q3L ≡  d3L  ∼ (3, 3, 2/3) , (1.1) J3L uaR ∼ (3, 1, 2/3) , daR ∼ (3, 1, −1/3) , JαR ∼ (3, 1, −4/3) , J3R ∼ (3, 1, 5/3) , (1.2) where a = 1, 2, 3 and α = 1, 2 are family indices. The quantum numbers in parentheses are given upon 3-3-1 symmetries,√respec- tively. The electric charge operator takes the form Q = T3 − 3T8 +X , where Ti (i = 1, 2, ..., 8) are SU (3)L charges, while X is that of U (1)X . The new quarks possess exotic electric charges as Q(Jα ) = −4/3 and Q(J3 ) = 5/3.
11 The model can work with only two scalar triplets: η10 χ−     1 − −− η =  η2  ∼ (1, 3, 0), χ =  χ2  ∼ (1, 3, −1), (1.3) η3+ χ03 with VEVs     u 0 1  1  hηi = √ 0 , hχi = √ 0 . (1.4) 2 0 2 w 1.2 Scalar sector The scalar potential of the model is given by: Vsimple (χ, η) = µ21 η † η + µ22 χ† χ + λ1 (η † η)2 + λ2 (χ† χ)2 + λ3 (η † η)(χ† χ) + λ4 (η † χ)(χ† η), (1.5) where µ1,2 have mass-dimensions while λ1,2,3,4 are dimensionless. Expanding η, χ around the VEVs, we get u S1 √ +iA1 χ−         √ 2 2 0 1 η = 0 + η2− , χ= 0 + χ−− 2 , (1.6) w S3 √ +iA3 0 η3+ √ 2 2 The physical eigenstates are identified as follows h ≡ cξ S1 −sξ S3 , H ≡ sξ S1 + cξ S3 , H ± ≡ cθ η3± + sθ χ± 1 . The physical eigenvalues are cor- λ2 −λ2 respondingly m2h = λ1 u2 + λ2 w2 − (λ1 u2 − λ2 w2 )2 + λ23 u2 w2 ' 4λ12λ q 3 u2 , m2 = 2 H λ1 u2 + λ2 w2 + (λ1 u2 − λ2 w2 )2 + λ2 u2 w2 ' 2λ2 w2 , m2 ± = 4 (u2 + w2 ) ' 4 w2 . We have λ λ q 3 H 2 2 denoted cξ = cos(ξ), sξ = sin(ξ), cθ = cos(θ), sθ = sin θ. ξ and θ are alter- nately the mixing angles of S1 − S3 and χ1 − η3 . There are eight massless scalar fields GZ ≡ A1 , GZ 0 ≡ A3 , G± W ≡ η2 , G±± ± Y ≡ χ ±± 2 , G ± X ≡ cθ χ ± 1 − sθ η3 ± . In the effective limit, u w , we have G−    u+h+iGZ  √ X 2 η' G− , χ'  G−− Y .  (1.7) W w+H+iGZ 0 + √ H 2 1.3 Gauge sector The gauge bosons receive their masses from the following term of the Lagrangian, PΦ=η,χ (Dµ hΦi)† (Dµ hΦi), where Dµ = ∂µ + igs ti Giµ + igTi Aiµ +igX XBµ , with gs , g and gX are the gauge coupling constants,
12 while ti , Ti and X are respectively SU (3)C , SU (3)L and U (1)X charges; Giµ , Aiµ and Bµ are the gauge bosons, as associated with the 3-3-1 groups, respectively. The gluons Gi are massless and physical fields by themselves. The physical charged gauge bosons with masses are respectively given by A1 ∓ iA2 g2 2 W± ≡ √ , m2W = u , 2 4 A4 ∓ iA5 g2 2 X∓ ≡ √ , m2X = (w + u2 ), 2 4 A6 ∓ iA7 g2 2 Y ∓∓ ≡ √ , m2Y = w . (1.8) 2 4 Two physical neutral gauge bosons (beside the photon) with their masses identified by 1 2 g2 q m2Z1 = [mZ + m2Z 0 − (m2Z − m2Z 0 )2 + 4m4ZZ 0 ] ' 2 u2 , 2 4cW 1 2 g 2 c2W q m2Z2 = [mZ + m2Z 0 + (m2Z − m2Z 0 )2 + 4m4ZZ 0 ] ' w2(.1.9) 2 3(1 − 4s2W ) The model has shown that there is a mixing between two states, Z and Z 0 , with the mixing angle √ √ 3(1 − 4s2W )3/2 u2 3(1 − 4s2W )3/2 u2 t2ϕ = 4 2 2 ' , (1.10) 2cW w − (1 + 2sW )(1 − 4sW )u 2 2 2c4W w2 Because of ϕ 1, we have Z1 ' Z and Z2 ' Z 0 . The Z1 is the standard model like Z boson, while Z2 is a new neutral gauge boson with the mass in w scale. The contribution to the experimental ρ-parameter can be calculated as 2 m2W m4 0 1 − 4s2W u2 ∆ρ ≡ 2 2 − 1 ' 2 ZZ2 ' . (1.11) cW mZ1 mZ mZ 0 2c2W w2 Substituting s2W = 0.231 and ∆ρ < 0.0007, we have w > 460 GeV. Since the other constraints yield w in some TeV, we conclude that the ρ-parameter is very close to one and suitable to the experimen- tal data Note that if we choose a model with two scalars, chi and rho, then the rho parameter is too large to fit the experiment.
13 1.4 Fermion masses and proton stability To generate mass for fermions, we construct Yukawa interac- tions by two scalar triplets η and χ, LY = ¯ 3L χJ3R + hJαβ Q hJ33 Q ¯ αL χ∗ JβR u +hu3a Q ¯ 3L ηuaR + hαa Q ¯ αL ηχuaR Λ d +hdαa Q ¯ αL η ∗ daR + h3a Q ¯ 3L η ∗ χ∗ daR Λ h0e ¯c +heab ψ¯aL c ψbL η + ab (ψaL ηχ)(ψbL χ∗ ) Λ2 sν + ab (ψ¯aL c η ∗ )(ψbL η ∗ ) + H.c., (1.12) Λ where the Λ is a new scale (with the mass dimension) under which the effective interactions take place. heab is antisymmetric while sνab is symmetric in the flavor indices. The mass Lagrangian of quarks and charged leptons takes the form, −f¯aL√mfab fbR + H.c., wheref = J, u, d, e. The mass√of J3 is mJ33 = −hJ33 w/ 2. The mass matrix of J1,2 is mJαβ = −hJαβ w/ 2. They all get the masses propor- tional to w scale. The mass matrices of u, d and the charged lep- tons are respectively: mu3a = −hu3a √u2 , muαa = −huαa uw 2Λ ; mdαa = −hdαa √u2 , √ 2 md3a = hd3a uw 2Λ ; and meab = 2u heab + h0e ba 4Λ2 . Because of Λ ∼ w , they w all have the masses in the weak scale u = 246 GeV. For top quark, we havemt = −hu33 × 174 GeV leading to mt = 173 GeV if hu33 ≈ −1. The mass Lagrangian of neutrinos is given by − 12 ν¯aLc mνab νbL + H.c., where ν u2 mab = −sab Λ . In fact, using Λ = 5 TeV, u = 246 GeV and mνab ∼ eV, we ν have sνab = h ∼ 10−10 . The Yukawa coupling of electron is chosen h = he ∼ 10−6 , the lepton number violating parameter is obtained ∼ 10−4 . The strength of the violating interaction for approximate lepton number is reasonably small in comparison to the ordinary inter- actions, and this may be the reason that the neutrino masses are observed to be small. The neutrino mass matrix is symmetric and generalized.Hence it can be compared to the neutrino mixing angles and squared mass differences, as the independent analysis of the model.
14 1.5 FCNC The tree-level FCNCs are discribed by the Lagrangian, g ∗ 1 0 0 LFCNC = − p 2 (VqL )3i √ (VqL )3j q¯iL γ µ qjL Zµ0 (i 6= j), (1.13) 1 − 3tW 3 where we have denoted q as u either d. substituting Z 0 = −sϕ Z1 + cϕ Z2 , the effective Lagrangian for hadronic FCNCs can be derived via the Z1,2 exchanges as ! ∗ g 2 [(VqL )3i (VqL )3j ]2 s2ϕ c2ϕ 0 0 Leff FCNC = + 2 (¯ qiL γ µ qjL )2 . (1.14) 3(1 − 3t2W ) 2 mZ1 mZ2 The contribution of Z1 is negligible. Therefore, only Z2 governs the FCNCs and we have ∗ [(VqL )3i (VqL )3j ]2 0 µ 0 2 Leff FCNC ' (¯ qiL γ qjL ) . (1.15) w2 We can see that this interaction is independent of the Landau pole 1/(1 − 4s2W ) (this is also an evidence pointing out that when the theory is encountered with the Landau pole, the effective interac- tions take place). The strongest constraint comes from the K 0 − K¯ 0 ∗ 2 system, given by [(VdL )31w(V2 dL )32 ] < (104 1TeV)2 . Assume that ua is flavor-diagonal. The CKM matrix is just ∗ VdL (i.e., VCKM = VdL ). Therefore, |(VdL )31 (VdL )32 | ' 3.6 × 10−4 and we have w > 3.6 TeV. This limit is still in the perturbative region of the model and is suitable to the recent experimental bounds . By contrast, if the first or second generation of quarks is arranged differently from the two others under SU (3)L FCNCs will be large unreasonably.
15 Chapter 2 Inert scalar and dark matter In order to find candidates for DM, we consider alternately the rest of scalars (ρ, S), even the replications of η, χ as the inert sector (Z2 odd) responsible for dark matter. 2.1 Simple 3-3-1 model with inert ρ triplet We can introduce into the theory constructed above an extra scalar triplet which transforms as an odd field under a Z2 symme- ++ T try, as follows ρ = ρ+ 0 ∼ (1, 3, 1). The normal scalar sector 1 , ρ2 , ρ3 (η, χ) which consists of the VEVs, the conditions for parameters and the physical scalars with their masses as obtained above re- mains unchanged . For the inert sector, ρ has vanishing VEVs due to the Z2 conservation. Moreover, the real and imaginary parts of electrically-neutral complex field ρ02 = √12 (Hρ + iAρ ) by themselves are physical fields. Any one of them can be stabilized if it is the lightest inert particle (LIP) among the inert particles resided in ρ due to the Z2 symmetry. we can show that Hρ and Aρ cannot be a dark matter. Indeed, Hρ and Aρ are not separated (degenerate) in mass which leads to a scattering cross-section of Hρ and Aρ off nuclei due to the t-channel exchange by Z boson. Such a large contribution has already been ruled out by the direct dark matter detection experiments. 2.2 Simple 3-3-1 model with η replication The second hypothesis is that the model is added to the η repli- 0 00 0− 0+ T cation defined by η = η1 , η2 , η3 ∼ (1, 3, 0). Here, the η 0 and η
16 have the same gauge quantum numbers. The η0 is assigned as an odd field under the Z2 , η0 → −η0 , whereas the η and all other fields of the simple 3-3-1 model are even. The scalar potential that is invariant under the gauge symmetry and Z2 is given by V = Vsimple + µ2η0 η 0† η 0 + x1 (η 0† η 0 )2 + x2 (η † η)(η 0† η 0 ) + x3 (χ† χ)(η 0† η 0 ) 1 +x4 (η † η 0 )(η 0† η) + x5 (χ† η 0 )(η 0† χ) + [x6 (η 0† η)2 + H.c.], (2.1) 2 where µη0 has the dimension of mass while xi (i = 1, 2, 3, ..., 6) are dimensionless. The physical inert particle masses are determined by 1 1 m2H10 = Mη20 + (x4 + x6 )u2 , m2A01 = Mη20 + (x4 − x6 )u2 , 2 2 1 m2η20 = Mη20 , m2η30 = Mη20 + x5 w 2 , (2.2) 2 where Mη20 ≡ µ2η0 + 12 x2 u2 + 21 x3 w2 and η100 ≡ √12 (H10 + iA01 ). The lightest inert particle (LIP) responsible for DM is H10 if x6 < Min{0, −x4 , (w/u)2 x5 − x4 }. Or alternatively A01 if x6 > Max{0, x4 , x4 − (w/u)2 x5 }. Let us consider the case H10 as the dark matter candidate (or a LIP). The H10 transforms as a doublet dark matter under the standard model symmetry. 2.3 Simple 3-3-1 model with χ replication T We introduce the χ replication (Z2 odd), χ0 = χ0− 0−− , χ00 1 , χ2 3 ∼ (1, 3, −1). The scalar potential that is invariant under the gauge symmetry and Z2 is given by: V = Vsimple + µ2χ0 χ0† χ0 + y1 (χ0† χ0 )2 + y2 (η † η)(χ0† χ0 ) + y3 (χ† χ)(χ0† χ0 ) 1 +y4 (η † χ0 )(χ0† η) + y5 (χ† χ0 )(χ0† χ) + [y6 (χ0† χ)2 + H.c.], (2.3) 2 0 where χ00 0 0 3 ≡ √2 (H3 + iA3 ).Depending on the parameter regime, H3 1 0 or A3 may be the LIP responsible for dark matter. Let us consider H30 as the LIP. The H30 is a singlet dark matter under the standard model symmetry. 2.4 Simple 3-3-1 model with inert scalar sextet We consider two cases in which two inert scalar sextet with X = 0 and X = 1 are introduced respectively.
17 2.4.1 Inert scalar sextet X = 0 The inert sextet with X = 0 has the following form − +   0 S12 S13 S11 √ 2 √ 2 −  0  S12 −− S23 S=  ∼ (1, 6, 0). (2.4)   √ 2 S22 √ 2 +  0  S13 S23 ++ √ 2 √ 2 S33 This sextet is odd under the Z2 (S → −S), whereas all the other fields are even. The scalar potential that is invariant under the gauge symmetry and Z2 is given by V = Vsimple + µ2S TrS † S + z1 (TrS † S)2 + z2 Tr(S † S)2 +(z3 η † η + z4 χ† χ)TrS † S + z5 η † SS † η + z6 χ† SS † χ 1 + (z7 ηηSS + H.c.), (2.5) 2 Depending on the parameter space, HS , AS , HS0 and A0S may be dark matter candidates. However, HS andAS are similar to Hρ and Aρ which has already been ruled out by the direct dark matter detection experiments. By contrast,HS0 and A0S transform as dou- blets under the standard model symmetry and are separated in the masses, but they cannot be the LIP because both are much heavier than the H1 field. Therefore, they will rapidly decay that cannot be dark matter. To conclude, the scalar sextet S with X = 0 does not provide realistic dark matter candidates. 2.4.2 Inert scalar sextet X = 1 Let us introduce another sextet with X = 1 (Z2 odd), ++  0  + σ12 σ13 σ11 √ 2 √ 2  0 +  σ12 − σ23 σ=  ∼ (1, 6, 1). (2.6)   √ 2 σ22 √ 2 ++ +   σ13 σ23 +++ √ 2 √ 2 σ33 The scalar potential is given by V = Vsimple + µ2σ Trσ † σ + t1 (Trσ † σ)2 + t2 Tr(σ † σ)2 +(t3 η † η + t4 χ† χ)Trσ † σ + t5 η † σσ † η + t6 χ† σσ † χ 1 + (t7 χχσσ + H.c.). (2.7) 2 Here, we obtain either the Hσ or the Aσ can be regarded as the LIP responsible for dark matter. Without lost of generality, in the following let us consider Hσ as the dark matter candidate.
18 2.5 An evaluation of dark matter observables To be concrete, in the following we present for the case of the sextet dark matter (Hσ ). There are various channels that might contribute to the relic density such as Hσ Hσ → hh, ttc , W + W − , ZZ, as well as the annihilations → ZW ± , AW ± , t±2/3 b±1/3 and H H → hh, ttc , W + W − , ZZ, ZA, AA. They ± ± ∓ Hσ H 1 1 1 are given by the diagrams in Fig. 2.1 and Fig. 2.2 with respect to the Higgs and gauge portals, respectively. The annihilation cross-section times relative velocity is defined as Hσ (H1+) h Hσ (H1+ ) tc h Hσ (H1−) h Hσ (H1− ) t Hσ (H1+) h Hσ (H1+ ) W +, Z h h Hσ (H1−) h Hσ (H1− ) W −, Z Hσ h H1+ h Hσ H1 Hσ h H1− h Figure 2.1: Contributions to Hσ or H1± annihilation via the Higgs portal when they are lighter than the new particles of the simple 3-3-1 model. There are additionally two u-channels that can be derived from the corresponding t-channels above. " 2 2 # α2 2.3 TeV λ × 0.782 TeV hσvi ' + , (2.8) (150 GeV)2 mHσ mHσ where λ ≡ t3 + t5 /2, and α = 1/128. Note also that the quantity α2 /(150 GeV)2 ' 1 pb has been factorized fora further convenience. The relic density can fit the data by this case if 0.1pb Ωh2 ' ' 0.11, (2.9) hσvi